Optimal. Leaf size=264 \[ \frac {12 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^5 (a+b x)}-\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{5 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{3 e^5 (a+b x)}+\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^5 (a+b x)}-\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{9 e^5 (a+b x)} \]
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Rubi [A] time = 0.10, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {770, 21, 43} \begin {gather*} \frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^5 (a+b x)}-\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{9 e^5 (a+b x)}+\frac {12 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^5 (a+b x)}-\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{5 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{3 e^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 770
Rubi steps
\begin {align*} \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^3 \sqrt {d+e x} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^4 \sqrt {d+e x} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^4 \sqrt {d+e x}}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^{3/2}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{5/2}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{7/2}}{e^4}+\frac {b^4 (d+e x)^{9/2}}{e^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}-\frac {8 b (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}-\frac {8 b^3 (b d-a e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac {2 b^4 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 172, normalized size = 0.65 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{3/2} \left (1155 a^4 e^4+924 a^3 b e^3 (3 e x-2 d)+198 a^2 b^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+44 a b^3 e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 e^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 34.98, size = 241, normalized size = 0.91 \begin {gather*} \frac {2 (d+e x)^{3/2} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (1155 a^4 e^4+2772 a^3 b e^3 (d+e x)-4620 a^3 b d e^3+6930 a^2 b^2 d^2 e^2+2970 a^2 b^2 e^2 (d+e x)^2-8316 a^2 b^2 d e^2 (d+e x)-4620 a b^3 d^3 e+8316 a b^3 d^2 e (d+e x)+1540 a b^3 e (d+e x)^3-5940 a b^3 d e (d+e x)^2+1155 b^4 d^4-2772 b^4 d^3 (d+e x)+2970 b^4 d^2 (d+e x)^2+315 b^4 (d+e x)^4-1540 b^4 d (d+e x)^3\right )}{3465 e^4 (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 245, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (315 \, b^{4} e^{5} x^{5} + 128 \, b^{4} d^{5} - 704 \, a b^{3} d^{4} e + 1584 \, a^{2} b^{2} d^{3} e^{2} - 1848 \, a^{3} b d^{2} e^{3} + 1155 \, a^{4} d e^{4} + 35 \, {\left (b^{4} d e^{4} + 44 \, a b^{3} e^{5}\right )} x^{4} - 10 \, {\left (4 \, b^{4} d^{2} e^{3} - 22 \, a b^{3} d e^{4} - 297 \, a^{2} b^{2} e^{5}\right )} x^{3} + 6 \, {\left (8 \, b^{4} d^{3} e^{2} - 44 \, a b^{3} d^{2} e^{3} + 99 \, a^{2} b^{2} d e^{4} + 462 \, a^{3} b e^{5}\right )} x^{2} - {\left (64 \, b^{4} d^{4} e - 352 \, a b^{3} d^{3} e^{2} + 792 \, a^{2} b^{2} d^{2} e^{3} - 924 \, a^{3} b d e^{4} - 1155 \, a^{4} e^{5}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 556, normalized size = 2.11 \begin {gather*} \frac {2}{3465} \, {\left (4620 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} b d e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 1386 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2} b^{2} d e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 396 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a b^{3} d e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 11 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} b^{4} d e^{\left (-4\right )} \mathrm {sgn}\left (b x + a\right ) + 924 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{3} b e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 594 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{2} b^{2} e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 44 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a b^{3} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} b^{4} e^{\left (-4\right )} \mathrm {sgn}\left (b x + a\right ) + 3465 \, \sqrt {x e + d} a^{4} d \mathrm {sgn}\left (b x + a\right ) + 1155 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 202, normalized size = 0.77 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (315 b^{4} e^{4} x^{4}+1540 a \,b^{3} e^{4} x^{3}-280 b^{4} d \,e^{3} x^{3}+2970 a^{2} b^{2} e^{4} x^{2}-1320 a \,b^{3} d \,e^{3} x^{2}+240 b^{4} d^{2} e^{2} x^{2}+2772 a^{3} b \,e^{4} x -2376 a^{2} b^{2} d \,e^{3} x +1056 a \,b^{3} d^{2} e^{2} x -192 b^{4} d^{3} e x +1155 a^{4} e^{4}-1848 a^{3} b d \,e^{3}+1584 a^{2} b^{2} d^{2} e^{2}-704 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{3465 \left (b x +a \right )^{3} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 384, normalized size = 1.45 \begin {gather*} \frac {2 \, {\left (35 \, b^{3} e^{4} x^{4} - 16 \, b^{3} d^{4} + 72 \, a b^{2} d^{3} e - 126 \, a^{2} b d^{2} e^{2} + 105 \, a^{3} d e^{3} + 5 \, {\left (b^{3} d e^{3} + 27 \, a b^{2} e^{4}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{2} e^{2} - 9 \, a b^{2} d e^{3} - 63 \, a^{2} b e^{4}\right )} x^{2} + {\left (8 \, b^{3} d^{3} e - 36 \, a b^{2} d^{2} e^{2} + 63 \, a^{2} b d e^{3} + 105 \, a^{3} e^{4}\right )} x\right )} \sqrt {e x + d} a}{315 \, e^{4}} + \frac {2 \, {\left (315 \, b^{3} e^{5} x^{5} + 128 \, b^{3} d^{5} - 528 \, a b^{2} d^{4} e + 792 \, a^{2} b d^{3} e^{2} - 462 \, a^{3} d^{2} e^{3} + 35 \, {\left (b^{3} d e^{4} + 33 \, a b^{2} e^{5}\right )} x^{4} - 5 \, {\left (8 \, b^{3} d^{2} e^{3} - 33 \, a b^{2} d e^{4} - 297 \, a^{2} b e^{5}\right )} x^{3} + 3 \, {\left (16 \, b^{3} d^{3} e^{2} - 66 \, a b^{2} d^{2} e^{3} + 99 \, a^{2} b d e^{4} + 231 \, a^{3} e^{5}\right )} x^{2} - {\left (64 \, b^{3} d^{4} e - 264 \, a b^{2} d^{3} e^{2} + 396 \, a^{2} b d^{2} e^{3} - 231 \, a^{3} d e^{4}\right )} x\right )} \sqrt {e x + d} b}{3465 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (a+b\,x\right )\,\sqrt {d+e\,x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x\right ) \sqrt {d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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